Homology Classes of 3-∆-complexes Made up of a Small Number of Simplexes V. Nardozza∗ September 2014y Abstract By means of a computer, all the possible homogeneous compact 3-∆- complexes made up of a small number of simplexes (from 1 to 3) have been classified in homology classes. The analysis shows that, with a small number of simplexes, it is already possible to build quite a large number of separate topological spaces. Key Words: topology, 3-delta-complexes. 1 Introduction Given a number T of 3-simplexes, having n = 4 · T faces, it is possible to build several compact topological spaces by identifying the n faces in couples. There are: (n − 1) · (n − 3) · ::: · 1 (1) different combinations possible for identifying faces and, for each couple of faces, there are 6 different orientations for the identification. In total, all the possible compact 3-∆-complexes that is possible to build up with a number T of 3-simplexes (i.e. 4 · T faces) are: n (n − 1) · (n − 3) · ::: · 1 · 6 2 (2) where the above equation takes already into account the most obvious symme- tries. For example, with 1 tetrahedron is possible to build 108 different compact ∆- complexes, with 2 tetrahedra is possible to build 136080 compact ∆-complexes and so on. Note that some of the combination lead to non feasible ∆-complexes meaning that, given the instruction for identifying faces, it is simply not possible to build a space that makes sense. By means of a computer we have evaluated the homology groups of all the possible ∆-complexes made up of T simplexes (with T going from 1 to 3) and we have classified them is classes of complexes having the same homology group. The homology groups have been evaluated from the boundary maps @i ex- pressed as a matrix Di of integers. In particular the rank of the homology ∗Electronic Engineer (MSc). Turin, Italy. mailto: [email protected] yPosted at: www.vixra.org in the section Topology 1 group Hi has been evaluated from the ranks of Di+1;Di, and the chain space dimensions while the torsion part of the group Hi has been red out from the boundary map Di+1 expressed in Smith normal form. The results are reported below. Type of comp. Num. of comp. Note Non feasible complexes 69 Path connected complexes 39 in 8 homology classes Non path connected complexes 0 Total 108 Table 1 : Complexes made of 1 simplex Type of comp. Num. of comp. Note Non feasible complexes 98991 Path connected complexes 35568 in 50 homology classes Non path connected complexes 1521 Total 136080 Table 2 : Complexes made of 2 simplexes Type of comp. Num. of comp. Note Non feasible complexes 366924249 Path connected complexes 113844096 in 234 homology classes Non path connected complexes 4220775 Total 484989120 Table 3 : Complexes made of 3 simplexes 2 Spaces made of 1 simplex - sum up table The following table sums up the homology classes of all path connected compact homogeneous 3-∆-complexes made up using only of 1 simplex. n H0 H1 H2 H3 χ Num. of Compl. 1 Z 0 0 Z 0 20 2 Z Z4 0 Z 0 5 3 Z Z5 0 Z 0 5 4 Z 0 Z Z 1 3 5 Z 0 0 0 1 3 6 Z 0 Z 0 2 1 7 Z Z4 0 0 1 1 8 Z Z5 0 0 1 1 Table 4 : Homology classes (spaces made of 1 simplex) 2 3 Spaces made of 2 simplexes - sum up table The following table sums up the homology classes of of all path connected compact homogeneous 3-∆-complexes made up using only of 2 simplexes. n H0 H1 H2 H3 χ Num. of Compl. 1 Z 0 Z Z2 0 11359 2 Z 0 0 Z2 -1 6473 3 Z 0 0 Z 0 5621 4 Z 0 Z Z 1 3226 5 Z Z3 0 Z 0 1765 6 Z 0 0 0 1 1595 7 Z 0 Z 0 2 903 8 Z Z2 0 Z 0 732 9 Z Z3 0 0 1 578 10 Z 0 Z2 Z2 1 440 11 Z Z 0 Z -1 396 12 Z Z7 0 Z 0 364 13 Z Z2 0 0 1 358 14 Z Z5 0 Z 0 212 15 Z Z8 0 Z 0 180 16 Z Z7 0 0 1 165 17 Z 0 Z2 0 3 154 18 Z Z3 Z 0 2 142 19 Z 0 Z2 0 1 139 20 Z Z 0 0 0 130 2 21 Z Z5 0 Z -1 114 22 Z Z2 Z 0 2 94 23 Z Z8 0 0 1 87 24 Z Z5 0 0 1 83 25 Z Z7 Z 0 2 40 Table 5 : Homology classes (spaces made of 2 simplexes) n:1-25 3 n H0 H1 H2 H3 χ Num. of Compl. 26 Z Z Z 0 1 32 27 Z Z2 Z Z 1 25 28 Z Z2 ⊕ Z2 0 Z 0 22 29 Z Z2 ⊕ Z2 0 0 1 21 30 Z Z5 Z 0 2 20 31 Z Z8 Z 0 2 14 32 Z Z Z Z 0 12 33 Z 0 Z3 0 4 11 2 34 Z Z2 Z 0 3 11 35 Z Z3 Z Z 1 8 36 Z Z7 Z Z 1 5 37 Z 0 Z2 Z 2 4 38 Z Z2 Z2 0 1 4 39 Z Z2 ⊕ Z2 Z 0 2 4 40 Z Z8 Z Z 1 4 41 Z Z Z2 0 2 4 42 Z 0 Z4 0 5 3 2 43 Z Z8 Z 0 3 3 2 44 Z Z3 Z 0 3 2 45 Z Z5 Z Z 1 2 2 46 Z Z7 Z 0 3 2 47 Z Z Z2 0 0 2 2 48 Z Z2 ⊕ Z2 Z 0 3 1 3 49 Z Z3 Z 0 4 1 2 50 Z Z5 Z 0 3 1 Table 5 : Homology classes (spaces made of 2 simplexes) n:26-50 4 Spaces made of 3 simplexes - sum up table The following table sums up the homology classes of of all path connected compact homogeneous 3-∆-complexes made up using only of 3 simplexes. 4 n H0 H1 H2 H3 χ Num. of Compl. 1 Z 0 Z Z3 -1 27794856 2 Z 0 0 Z2 -1 21297916 3 Z 0 Z Z2 0 14583893 4 Z 0 0 Z3 -2 13190400 5 Z 0 0 Z 0 8256514 6 Z 0 Z Z 1 7289062 7 Z 0 Z 0 2 3337122 8 Z 0 0 0 1 1952888 9 Z 0 Z2 0 3 1664641 10 Z 0 Z2 Z 2 1573652 11 Z 0 Z2 Z3 0 1464514 12 Z 0 Z2 Z2 1 1350393 13 Z Z2 0 Z 0 1039055 14 Z 0 Z2 0 1 777592 15 Z Z2 Z Z 1 648812 2 16 Z Z2 Z Z 0 612947 17 Z Z2 0 0 1 591869 18 Z Z3 0 Z 0 492526 19 Z 0 Z ⊕ Z2 0 2 462966 2 20 Z Z3 0 Z -1 453676 21 Z Z2 Z 0 2 445639 22 Z 0 Z3 0 4 417638 2 23 Z Z2 0 Z -1 369919 2 24 Z Z4 0 Z -1 338930 2 25 Z Z5 0 Z -1 295738 26 Z Z5 0 Z 0 236229 27 Z Z4 0 Z 0 226886 28 Z Z3 0 0 1 226760 29 Z 0 Z2 Z 0 163893 30 Z Z3 Z 0 2 144341 3 31 Z Z4 0 Z -2 141960 3 32 Z Z5 0 Z -2 141960 33 Z 0 Z3 Z 3 125687 2 34 Z Z2 Z 0 3 115262 35 Z Z3 Z Z 1 86689 36 Z Z2 Z2 0 1 86288 37 Z Z5 0 0 1 81232 2 38 Z Z2 Z Z 2 74244 39 Z Z4 0 0 1 69779 40 Z 0 Z ⊕ Z2 Z 1 68910 Table 6 : Homology classes (spaces made of 3 simplexes) n:1-40 5 n H0 H1 H2 H3 χ Num. of Compl. 41 Z 0 Z4 0 5 68511 42 Z Z 0 Z -1 64013 43 Z Z 0 Z2 -2 43511 2 44 Z Z3 Z 0 3 39050 45 Z Z4 Z Z 1 39043 46 Z Z5 Z Z 1 38081 47 Z Z7 0 Z 0 37748 48 Z Z 0 0 0 37237 49 Z 0 Z3 0 1 35734 2 50 Z Z3 Z Z 0 34861 51 Z Z3 Z2 0 1 30964 52 Z Z5 Z 0 2 29680 53 Z Z7 0 0 1 28507 2 2 54 Z Z2 Z Z 1 27856 55 Z Z4 Z 0 2 26523 56 Z Z Z 0 1 24640 57 Z Z6 0 Z 0 23764 58 Z Z6 0 0 1 21663 59 Z Z9 0 Z 0 19200 60 Z Z11 0 Z 0 19180 3 61 Z Z2 Z 0 4 18905 62 Z Z6 Z 0 2 17551 63 Z Z8 0 Z 0 16746 64 Z Z8 0 0 1 16650 65 Z 0 Z3 Z2 2 15037 66 Z Z ⊕ Z2 Z 0 1 13680 67 Z Z11 0 0 1 13321 68 Z Z9 0 0 1 13235 69 Z Z Z Z 0 11151 70 Z Z7 Z 0 2 10450 71 Z Z10 0 Z 0 9654 72 Z Z12 0 Z 0 9536 73 Z Z13 0 Z 0 9528 74 Z Z6 Z2 0 1 9410 75 Z 0 Z4 Z 4 8774 76 Z Z Z2 0 0 8684 77 Z Z ⊕ Z2 Z2 0 0 8441 78 Z Z2 ⊕ Z2 0 0 1 8176 79 Z Z ⊕ Z2 0 0 0 7947 80 Z 0 Z5 0 6 7444 Table 6 : Homology classes (spaces made of 3 simplexes) n:41-80 6 n H0 H1 H2 H3 χ Num.